Let a- - b- - c- be three vectors such that a- 0 and a-b- - 2 a-c- - - a- - - - c- - - 1 - - b- - - 4 and - b-c- - - - 15 - If b- - 2 c- - -a- then - is equal to

The correct option is D 4 Given: a⃗, b⃗, c⃗ are vectors s.t. a⃗≠ 0, a⃗×b⃗=2a⃗×c⃗ |a⃗|=|c⃗|=1, |b⃗|=4, |b⃗×c⃗|=√(1⃗5) . b⃗ 2c⃗=λa⃗ To f ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

Let a⃗ , b⃗...

Question

Let $\to a, \to b, \to c$ be three vectors such that $\to a \neq 0$ and $\to a \times \to b = 2 \to a \times \to c, | \to a | = | \to c | = 1, | \to b | = 4$ and $| \to b \times \to c | = \sqrt{15}$ . If $\to b - 2 \to c = λ \to a$ , then $λ$ is equal to

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Solution

The correct option is D $- 4$
Given:
$\to a, \to b, \to c$ are vectors $s . t . \to a \neq 0, \to a \times \to b = 2 \to a \times \to c$
$| \to a | = | \to c | = 1, | \to b | = 4, | \to b \times \to c | = \sqrt{\to 1 5}$ .
$\to b - 2 \to c = λ \to a$
To find : $λ$
$\to b - 2 \to c = λ \to a$
$∴ | \to b - 2 \to c | = | λ | | \to a |$
$∴ \sqrt{| \to b n |^{2} + | 2 \to c |^{2} - 4 | \to b | | \to c | cos θ} = | λ | | \to a | - - - (1)$
$| \to a \times \to c | = \sqrt{15} . . . . . .$ (From given data
$∴ | \to b | | \to c | . sin θ \sqrt{15} \Rightarrow sin θ = \frac{\sqrt{15}}{4}$
$∴ cos θ = \frac{1}{4} . . . . . . (2)$
$∴ \sqrt{16 + 4 - 4.4 \times \frac{1}{4}} = | λ | | \to a |$
$∴ \sqrt{16} = | λ | .1$
$∴ | λ | = 4$
$| λ | = 4$
$B . - 4$

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Similar questions

Let $\to a, \to b, \to c$ be three vectors such that $| \to a | = | \to c | = 1; | \to b | = 4$ and $| \to b \times \to c | = \sqrt{15}$ . If $\to b - 2 \to c = λ \to a$ then a value of $λ$ is

Q. If

\to A, \to B

and

\to C

are vectors such that

| \to B | = | \to C |

, prove that

[(\to A + \to B) \times (\to A + \to C)] \times (\to B \times \to C) . (\to B + \to C) = 0

Q. If

\to a + 2 \to b + 3 \to c = 0

and

\to a \times \to b + \to b \times \to c + \to c \times \to a

is equal to

λ (\to b \times \to c)

then

λ

is equal to

Q. If

\to a

and

\to b

are perpendicular unit vectors and vector

\to c

is such that

\to c = \to a + \to b

, then

(\to a \times \to b) . (\to b \times \to c) + (\to b \times \to c) . (\to c \times \to a) + (\to c \times \to a) . (\to a \times \to b)

Q. If

\to a, \to b

and

\to c

are perpendicular to

\to b + \to c, \to c + \to a

and

\to a + \to b

respectively and if

| \to a + \to b | = 6, | \to b + \to c | = 8

and

| \to c + \to a | = 10

, the

| \to a + \to b + \to c |

is equal to

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Let →a,→b,→c be three vectors such that →a≠0 and →a×→b=2→a×→c,|→a|=|→c|=1,|→b|=4 and |→b×→c|=√15. If →b−2→c=λ→a, then λ is equal to

Let $\to a, \to b, \to c$ be three vectors such that $\to a \neq 0$ and $\to a \times \to b = 2 \to a \times \to c, | \to a | = | \to c | = 1, | \to b | = 4$ and $| \to b \times \to c | = \sqrt{15}$ . If $\to b - 2 \to c = λ \to a$ , then $λ$ is equal to